By paving the whole \(\mathbb {Z}^{2}\)  with such equations, we
get a partial difference equation which can be in
principle studied using known methods.
Since a priori
\(Q\ne |Q \ne \underline{Q} \ne |\underline{Q}\) ,
the obtained lattice will be a four stripe lattice, i.e., an extension of the Black and White
lattice considered in [1]}, [2]}, [3]}. This gives rise to lattice equations with two-periodic
coefficients for an unknown function \(u_{n,m}\) , with \((n,m)\in \mathbb {Z}^{2}\)
\(F^{(+)}_{n}F^{(+)}_{m}Q(u_{n,m},u_{n+1,m},u_{n,m+1},u_{n+1,m+1};\alpha _{1},\alpha _{2})\nonumber \\\qquad {}+F^{(-)}_{n}F^{(+)}_{m}|Q (u_{n,m},u_{n+1,m},u_{n,m+1},u_{n+1,m+1};\alpha _{1},\alpha _{2})\nonumber \\\qquad {}+F^{(+)}_{n}F^{(-)}_{m}\underline{Q} (u_{n,m},u_{n+1,m},u_{n,m+1},u_{n+1,m+1};\alpha _{1},\alpha _{2})\nonumber \\\qquad {}+F^{(-)}_{n}F^{(-)}_{m}|\underline{Q} (u_{n,m},u_{n+1,m},u_{n,m+1},u_{n+1,m+1};\alpha _{1},\alpha _{2}) =0, \) 
